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Theorem brafnmul 22586
Description: Anti-linearity property of bra functional for multiplication. (Contributed by NM, 31-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
brafnmul  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( bra `  ( A  .h  B )
)  =  ( ( * `  A ) 
.fn  ( bra `  B
) ) )

Proof of Theorem brafnmul
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hvmulcl 21648 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
2 brafval 22578 . . 3  |-  ( ( A  .h  B )  e.  ~H  ->  ( bra `  ( A  .h  B ) )  =  ( x  e.  ~H  |->  ( x  .ih  ( A  .h  B ) ) ) )
31, 2syl 15 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( bra `  ( A  .h  B )
)  =  ( x  e.  ~H  |->  ( x 
.ih  ( A  .h  B ) ) ) )
4 cjcl 11637 . . . 4  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
5 brafn 22582 . . . 4  |-  ( B  e.  ~H  ->  ( bra `  B ) : ~H --> CC )
6 hfmmval 22374 . . . 4  |-  ( ( ( * `  A
)  e.  CC  /\  ( bra `  B ) : ~H --> CC )  ->  ( ( * `
 A )  .fn  ( bra `  B ) )  =  ( x  e.  ~H  |->  ( ( * `  A )  x.  ( ( bra `  B ) `  x
) ) ) )
74, 5, 6syl2an 463 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( * `  A )  .fn  ( bra `  B ) )  =  ( x  e. 
~H  |->  ( ( * `
 A )  x.  ( ( bra `  B
) `  x )
) ) )
8 his5 21720 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  ~H  /\  B  e.  ~H )  ->  (
x  .ih  ( A  .h  B ) )  =  ( ( * `  A )  x.  (
x  .ih  B )
) )
983expa 1151 . . . . . 6  |-  ( ( ( A  e.  CC  /\  x  e.  ~H )  /\  B  e.  ~H )  ->  ( x  .ih  ( A  .h  B
) )  =  ( ( * `  A
)  x.  ( x 
.ih  B ) ) )
109an32s 779 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  x  e.  ~H )  ->  ( x  .ih  ( A  .h  B
) )  =  ( ( * `  A
)  x.  ( x 
.ih  B ) ) )
11 braval 22579 . . . . . . 7  |-  ( ( B  e.  ~H  /\  x  e.  ~H )  ->  ( ( bra `  B
) `  x )  =  ( x  .ih  B ) )
1211adantll 694 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  x  e.  ~H )  ->  ( ( bra `  B ) `  x
)  =  ( x 
.ih  B ) )
1312oveq2d 5916 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  x  e.  ~H )  ->  ( ( * `
 A )  x.  ( ( bra `  B
) `  x )
)  =  ( ( * `  A )  x.  ( x  .ih  B ) ) )
1410, 13eqtr4d 2351 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  x  e.  ~H )  ->  ( x  .ih  ( A  .h  B
) )  =  ( ( * `  A
)  x.  ( ( bra `  B ) `
 x ) ) )
1514mpteq2dva 4143 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( x  e.  ~H  |->  ( x  .ih  ( A  .h  B ) ) )  =  ( x  e.  ~H  |->  ( ( * `  A )  x.  ( ( bra `  B ) `  x
) ) ) )
167, 15eqtr4d 2351 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( * `  A )  .fn  ( bra `  B ) )  =  ( x  e. 
~H  |->  ( x  .ih  ( A  .h  B
) ) ) )
173, 16eqtr4d 2351 1  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( bra `  ( A  .h  B )
)  =  ( ( * `  A ) 
.fn  ( bra `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701    e. cmpt 4114   -->wf 5288   ` cfv 5292  (class class class)co 5900   CCcc 8780    x. cmul 8787   *ccj 11628   ~Hchil 21554    .h csm 21556    .ih csp 21557    .fn chft 21577   bracbr 21591
This theorem is referenced by:  cnvbramul  22750
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-hilex 21634  ax-hfvmul 21640  ax-hfi 21713  ax-his1 21716  ax-his3 21718
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-2 9849  df-cj 11631  df-re 11632  df-im 11633  df-hfmul 22369  df-bra 22485
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